# Diffraction Limit Discussion Continuation

Started Feb 21, 2014 | Discussions thread
Re: Um...
3

Jonny Boyd wrote:

Great Bustard wrote:

Jonny Boyd wrote:

Great Bustard wrote:

Jonny Boyd wrote:Diffraction causes a decrease in resolution, agreed?

Agreed.

When resolution drops due to stopping down from the peak aperture, that is due to diffraction, agreed?

Agreed.

At the aperture at which diffraction is reducing resolution, you can say that diffraction is limiting the resolution of the final image, agreed?

Agreed.

If resolution appears to be the same at an aperture smaller than the peak aperture then diffraction doesn't become the dominant factor in limiting resolution until later than the peak aperture, agreed?

Agreed.

Therefore, for practical purposes, as far as the eye can see, a system where resolution visibly drops immediately after peak aperture is more limited by diffraction than a system where the visible drop happens later. Agreed?

Not agreed, and am surprised you do not understand this. For example, let's say for a particular display size, viewing distance, and visual acuity, I can resolve 1000 lw/ph. All else equal, the photo from the lower MP sensor will dip below that threshold before the higher MP sensor.

What you're saying is that the lower resolution sensor produces lower resolution images. No kidding, that's what I've always said.

But how does this make the higher MP sensor more "diffraction limited" than the lower MP sensor?

As I've repeatedly explained, when diffraction produces a visible drop in resolution in situations where a lower resolution sensor wouldn't exhibit a visible drop in resolution, then diffraction is more a limiting factor for the image produced by the high res sensor. The high res sensor will still produce a higher resolution image, it's just that the resolution of that image is determined largely by diffraction effects, where as the lower resolution sensor's performance is dominated by the impact of its sensor and diffraction effects are relatively negligible.

My argument is that for a sufficiently low resolution sensor, an image taken at peak aperture and an image taken at a smaller aperture will have a difference in resolution that is indistinguishable to the naked eye because it is so minor. For practical purposes therefore the perceived resolution is not being limited by diffraction until an even smaller aperture than the actual peak.

But you can't see that the higher MP sensor had greater resolution at the peak aperture, so the higher MP sensor isn't being "visibly limited" by diffraction, either, until a much smaller aperture.

Of course it has higher resolution.

You're bringing up an interesting point though about whether the drop in resolution will be noticed for a high res sensor. To be honest, I hadn't really thought about things from that perspective, but that would be worth looking into, whether it is possible for the drop in resolution for a high resolution sensor and a high resolution lens being stopped down, is not visible.

In contrast, a higher resolution sensor will exhibit a drop in resolution immediately after the peak aperture which will be greater in relative and absolute terms and be more noticeable...

But it won't be noticeable -- that's the whole point. The higher MP sensor is resolving better than you can see, so you don't notice the drop in resolution.

Well the phase 'here MP sensor' is a relative one, comparing the resolution of two sensors, rather than comparing to the human eye, so a higher resolution sensor may not actually resolve better than you can see.

But I take your point that conceivably there could be a lens/camera combination where the changes could be undetectable because the resolution is so high. So Sensor resolution could have an impact at both the high end and the low end.

I suppose in this situation what you're really talking about is a system where the sensor effectively isn't limiting the resolution at all, just diffraction effects, so the question is then really whether the resolution of the lens at the next selectable aperture after the peak is sufficiently lower to be distinguished by the human eye. In essence you're taking the sensor out of the picture and asking whether the lens has a plateau instead of peak in terms of visible resolution changes. But even that would be determined by factors such as the physical size of the image, viewing, distance etc., so like with the low res sensor situation, in principle, under the right conditions, what you're saying is true. It's not trivial though to say when this would happen.

...therefore the system is limited by diffraction at an earlier aperture, while (as I have said on numerous occasions) having greater resolution than the lower resolution image.

No. Both the high and low MP sensors reached their peak at the same aperture,

Agreed.

and the higher MP sensor had higher resolution at every stop,

Agreed.

which was beyond what you could see.

Not necessarily. That would depend on the factors I mentioned above. To be fair, that's also true of the low res sensor situation. A difference that is imperceptible at a certain print size under certain viewing conditions would be visible in another situation.

Thus, if anything, the photo from the higher MP sensor dropped below that visible threshold at a smaller aperture than the photo from the lower MP sensor.

Under certain conditions, yes. Under other conditions, no.

I get the impression that some people see the words 'diffraction' and 'limit(ed)' in close proximity and freak out without reading what I've written.

At best what you're trying to say is that the higher MP sensor might go from, say, 3000 lw/ph at it's peak to 2000 lw/ph stopped down to some point, whereas the lower MP sensor might go from 2200 lw/ph to 1900 lw/ph at the same stopped down aperture.

That's hardly the best case scenario. It's entirely possible that the difference could be 3000–2000 vs. 200–199.9999

You are then arguing that we would notice a drop in resolution from 3000 lw/ph to 2000 lw/ph, but we wouldn't notice a drop in resolution from 2200 lw/ph to 1900 lw/ph. Thus, you conclude that the lens on the lower MP sensor is "diffraction limited" at a more narrow aperture than the lens on a higher MP sensor.

So, are you trying to define "diffraction limited" as when the lens resolution falls to a particular resolution of its peak value?

It's when the drop in resolution is visible. Which will depend on image size, viewing distance, etc.

I've done a few calculations to try and get more definitive numbers here, making use of the formula 1/r^2 = 1/l^2 + 1/s^2 where r is image resolution, l is lens resolution, and s is sensor resolution

If we take a lens with resolution at peak aperture of l_p, the width of the line pair in an image produced by a sensor will be w_p.

Similarly, at another aperture the resolution will be l_a and the width of a line pair in an image produced from the same sensor will be w_a.

We'll call the difference in these widths delta_w = w_a - w_p

At a distance of 1m, the acuity of a human eye allows it to resolve line pairs with separation of approximately w_0 = 0.3mm. In more general terms therefore, w_0 = 3 * 10^-4 * d.

If we took two images with a camera at peak aperture and another aperture, but which are otherwise identical, and viewed at a distance d, then at what point does it become impossible to tell that there has been drop in resolution? If the difference in the width of the line pairs at the two apertures in lower than the minimum line pair width that the eye can distinguish then the resolutions will appear to be the same.

If delta_w = w_a - w_p then we will perceive no difference in resolution when delta_w < w_0.

delta_w = w_a - w_p

= 1/r_a - 1/w_p

= sqrt (1/l_a^2 + 1/s^2) - sqrt (1/l_p^2) + 1/s^2)

If you choose appropriate units so that l_p =1, then you can plot graphs for how various sensor resolution (scaled relative to l_p) perform at apertures with various resolutions (scaled relative to l_p). Having chosen units so that l_p = 1, such graphs will give you delta_w * l_p and look like this:

I've chosen sensor resolutions that cover a useful range, from s = l_p x 10^-3 to s = l_p * 10^1. As you can see from the chart, this covers pretty much the entire range. Real world sensors will fall somewhere in between these values.

So where does delta_w become too small to notice a change in resolution? When delta_w < w_0. But we've charted delta_w / w_p, so we need to find:

delta_w < w_0

delta_w * l_p < w_0 * l_p

delta_w * l_p < 3 * 10^-4 * d * l_p

This so far assumes that the image is the same size as the sensor, wo now also need to include an enlargement factor e = h_i / h_s where h_i is the height of the image and h_s is the height of the sensor.

So what we need now is to find:

delta_w * l_p * e < 3 * 10^-4 * d * l_p

delta_w * l_p < 3 * 10^-4 * d / e * l_p

So now we need a chart of results for different values of d/e and l_p.

This shows w_0 * l_p for lenses of different resolutions (in lp/mm ) for different values of d/e (in mm)

So for example, an enlargement factor of 50, viewing distance of 50 mm, and a lens with resolution 1,000 lp/mm, w_0 * l_p = 0.3.

Looking at the first chart, this means that when the lens set to an aperture when the resolution is 0.7 times the peak, then delta_w < w_o for cameras with sensors of lower resolution than l_p.

So when the lens resolution is 1,000 lp/mm, the sensor resolution is lower than this, the viewing distance is 5cm and the enlargement factor is 50, then you have to stop down to an aperture with resolution less than 0.7 of the peak aperture to notice any drop in resolution due to diffraction. That's just one example.

How does that look?

Very pretty, lots of coloured lines.

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